4,776 research outputs found
Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations
This paper is concerned with the discretization error analysis of semilinear
Neumann boundary control problems in polygonal domains with pointwise
inequality constraints on the control. The approximations of the control are
piecewise constant functions. The state and adjoint state are discretized by
piecewise linear finite elements. In a postprocessing step approximations of
locally optimal controls of the continuous optimal control problem are
constructed by the projection of the respective discrete adjoint state.
Although the quality of the approximations is in general affected by corner
singularities a convergence order of is proven for domains
with interior angles smaller than using quasi-uniform meshes. For
larger interior angles mesh grading techniques are used to get the same order
of convergence
A FEM for an optimal control problem of fractional powers of elliptic operators
We study solution techniques for a linear-quadratic optimal control problem
involving fractional powers of elliptic operators. These fractional operators
can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic
problem posed on a semi-infinite cylinder in one more spatial dimension. Thus,
we consider an equivalent formulation with a nonuniformly elliptic operator as
state equation. The rapid decay of the solution to this problem suggests a
truncation that is suitable for numerical approximation. We discretize the
proposed truncated state equation using first degree tensor product finite
elements on anisotropic meshes. For the control problem we analyze two
approaches: one that is semi-discrete based on the so-called variational
approach, where the control is not discretized, and the other one is fully
discrete via the discretization of the control by piecewise constant functions.
For both approaches, we derive a priori error estimates with respect to the
degrees of freedom. Numerical experiments validate the derived error estimates
and reveal a competitive performance of anisotropic over quasi-uniform
refinement
A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domains
We propose and analyze a new discretization technique for a linear-quadratic
optimal control problem involving the fractional powers of a symmetric and
uniformly elliptic second oder operator; control constraints are considered.
Since these fractional operators can be realized as the Dirichlet-to-Neumann
map for a nonuniformly elliptic equation, we recast our problem as a
nonuniformly elliptic optimal control problem. The rapid decay of the solution
to this problem suggests a truncation that is suitable for numerical
approximation. We propose a fully discrete scheme that is based on piecewise
linear functions on quasi-uniform meshes to approximate the optimal control and
first-degree tensor product functions on anisotropic meshes for the optimal
state variable. We provide an a priori error analysis that relies on derived
Holder and Sobolev regularity estimates for the optimal variables and error
estimates for an scheme that approximates fractional diffusion on curved
domains; the latter being an extension of previous available results. The
analysis is valid in any dimension. We conclude by presenting some numerical
experiments that validate the derived error estimates
Optimal control of fractional semilinear PDEs
In this paper we consider the optimal control of semilinear fractional PDEs
with both spectral and integral fractional diffusion operators of order
with . We first prove the boundedness of solutions to both
semilinear fractional PDEs under minimal regularity assumptions on domain and
data. We next introduce an optimal growth condition on the nonlinearity to show
the Lipschitz continuity of the solution map for the semilinear elliptic
equations with respect to the data. We further apply our ideas to show
existence of solutions to optimal control problems with semilinear fractional
equations as constraints. Under the standard assumptions on the nonlinearity
(twice continuously differentiable) we derive the first and second order
optimality conditions
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